Constructing Mathematical Knowledge

Epistemology and Mathematical Education

Paul Ernest

• Publisher: Routledge
• Year: 2012
• eBook ISBN: 9781136364792
• Edition: 1
• Topic: Education
• Subtopic: Education General

First published in 1994. This book and its companion volume, Mathematics, Education and Philosophy: An International Perspective are edited collections. Instead of the sharply focused concerns of the research monograph, the books offer a panorama of complementary and forward-looking perspectives. They illustrate the breadth of theoretical and philosophical perspectives that can fruitfully be brough to bear on the mathematics and education. The empathise of this book is on epistemological issues, encompassing multiple perspectives on the learning of mathematics, as well as broader philosophical reflections on the genesis of knowledge. It explores constructivist and social theories of learning and discusses the rile of the computer in light of these theories.

Part 1

Constructivism and the Learning of Mathematics

We can know nothing that we have not made. (Vico, 1710, p. 76)

Until recently, mathematics has supported the deceptive appearance of being fully formed and perfectly finished knowledge. Hence only a few decades ago, the dominant view in mathematics education assumed that the teaching and learning of mathematics only required the effective transmission of mathematical knowledge. After all, did not Shannon and Weaver’s Communication Theory indicate that all that is necessary is that the message of mathematics be adequately coded, transmitted, received by the addressee, decoded and represented internally? Although the 1960s brought a softening of this view, with its emphasis on ‘discovery learning’, underpinning it was still an empiricist view of mind. Instead of receiving the message transmitted by the teacher, the learner now looked for mathematical knowledge in the world, albeit a carefully arranged and orchestrated corner of the world. Discovery learning also assumes that mathematical knowledge is pre-existing: it just needs to be actively noticed to be discovered. Furthermore, because it is ‘truth’, when discovered, it is recognized unproblematically.

The outlooks I have just described are not the products of naive error. To assert this would be false and unjust. Instead, they are a function of an empirical scientific or positivistic epistemology and methodological paradigm. This was the legacy of mathematics education, emerging from the disciplines of mathematics and experimental psychology. Even the influence of Piaget on the 1960s’ discovery learning approach was assimilated into this scientific research paradigm. However in the decade that followed a different paradigm emerged better reflecting Piaget’s psychology and his clinical interview method. Other strands combined with this, drawing on Kelly’s Personal Construct Theory, Problem-solving research from the Gestalt psychologists and early cognitive scientists such as Newell and Simon, as well as advances in cybernetics and other areas. The outcome was a new research paradigm for mathematics education, that of constructivism.

Piaget’s constructivism has its roots in an evolutionary biological metaphor, according to which the evolving organism must adapt to its environment in order to survive. Likewise, the developing human intelligence also undergoes a process of adaptation in order to fit with its circumstances and remain viable. Personal theories are constructed as constellations of concepts, and are adapted by the twin processes of assimilation and accommodation in order to fit with the human organism’s world of experience. Indeed Piaget claims that the human intelligence is ordering the very world it experiences in organizing its own cognitive structures. L’intelligence organise le monde en s’organisant elle-même (Piaget, 1937, cited in von Glasersfeld, 1989a, p. 162). Epistemologically, what is of tremendous significance in Piaget’s constructivism is (1) the notion that knowing is embodied, and essentially implicates interaction with the world, and (2) his ‘Genetic Epistemology’ that sees knowledge of the individual and group as historical and evolutionary, growing and changing to meet challenges and contradictions. This first aspect directly challenges the separation of mind and body in Cartesian Dualism that has so long dominated western thought. In their survey Varela et al. (1991) distinguish the three stages of ‘Cognitivism, Emergence and Enactive’ in the development of cognitive science, and locate Piaget’s contribution already in the last stage.

As part of the broader interdisciplinary movement of Structuralism, Piaget was seduced by the Bourbakian account of mathematics as logically constituted by three mother structures. Like many thinkers before him, Piaget afforded a privileged place to mathematical knowledge in his scheme. However, a scholar who has been influential in freeing constructivism from these constraints is von Glasersfeld. He has taken the scepticism of Piaget’s constructivism further, and argued in a radical version of constructivism that all knowledge, including mathematics, is constructed and fallible. That conclusion embroiled radical constructivism in a great deal of controversy in mathematics education, where until recently absolutist views of knowledge prevailed. However, the debate over the nature of mathematical knowledge, and in particular, that between absolutism and fallibilism, is not reflected in this present section on constructivist theories of the learning of mathematics.¹ For the contributors to this section, this debate is largely settled. Instead a new set of controversies has set in concerning such things as individual versus social construction. Epistemological issues, such as those raised by constructivism do seem to be dynamite in education!

Ultimately, the import of any theory of learning mathematics consists in facilitating interventions in the processes of its teaching and learning. Constructivism accounts for the individual idiosyncratic constructions of meaning, for systematic errors, misconceptions, and alternative conceptions in the learning of mathematics. Thus it facilitates diagnostic teaching, and the diagnosis and remediation of errors. A growing number of instructional projects is based on a constructivist sensitivity to children’s sense making and on the spontaneous strategies they have been observed to develop (Grouws, 1992). Until recently, the available literature on constructivist learning theory in mathematics has been limited to one or two monographs like Steffe et al. (1983). However, a growing number of publications is appearing offering a discussion of constructivist views on the learning and teaching of mathematics (e.g., Davis et al., 1990; Steffe, 1991; von Glasersfeld, 1991, 1994). In addition, there is a growing literature about the relationship between radical and individualistic versions of constructivism and social constructivism (e.g., Ernest, 1991, 1993; Steffe, in press). The present section adds to this literature, and begins to explore a number of significant issues for the future, such as the role of microcomputers in helping learners to construct mathematical knowledge and meaning. Microcomputers have great potential here, because they encourage children to think ‘outside their heads’, providing direct evidence of children’s learning and thought processes.

Note

1. But see the section on ‘Reconceptualizing the Philosophy of Mathematics’ in the companion volume Mathematics, Education and Philosophy.

References

Davis, R.B., Maher, C.A. and Noddings, N. (Eds) (1990) Constructivist Views on the Teaching and Learning of Mathematics (J.R.M.E. Monograph 4), Reston, Virginia, National Council of Teachers of Mathematics.

Ernest, P. (1991) The Philosophy of Mathematics Education, London, The Falmer Press.

Ernest, P. (1993) ‘Constructivism, the psychology of learning, and the nature of mathematics: Some critical issues’, Science and Education, 2, 2, pp. 87–93.

Grouws, D.A. (Ed) (1992) Handbook of Research on Mathematics Teaching and Learning, New York, Macmillan.

Steffe, L.P. (Ed) (1991) Epistemological Foundations of Mathematical Experience, New York, Springer-Verlag.

Steffe, L.P. and Gale, J. (Eds) (in press) Constructivism in Education, London, Erlbaum.

Steffe, L.P., von Glasersfeld, E., Richards, J. and Cobb, P. (1983) Children’s Counting Types: Philosophy, Theory, and Application, New York, Praeger.

Varela, F.J., Thompson, E. and Rosch, E. (1991) The Embodied Mind, Cambridge, Massachusetts, MIT Press.

Vico, G. (1710) De antiquissima Italorum sapientia, Nicolini, F. (Ed) Bari, Latzera (1914).

von Glasersfeld, E. (Ed) (1989a) ‘Constructivism in Education’, in Husen, T. (Ed) International Encyclopedia of Education (supplementary volume), Oxford, Pergamon.

von Glasersfeld, E. (Ed) (1991) Radical Constructivism in Mathematics Education, Dordrecht, Kluwer.

von Glasersfeld, E. (1994) Radical Constructivism: A Way of Knowing and Learning, London, The Falmer Press.

Chapter 1

A Radical Constructivist View of Basic Mathematical Concepts

Ernst von Glasersfeld

I am not a mathematician. My interest is in conceptual analysis, and mathematical concepts are extremely interesting — especially the seemingly simple ones that are linked to the basic elements of arithmetic. The most basic of these elements are the symbols that we call ‘numerals’.

There is an old statement that mathematics has to do with symbols and the manipulation of symbols. Frequent repetition of this statement has encouraged the belief that it is only the symbols that matter and that their conceptual referents need not be examined — presumably because to adults who have become used to doing arithmetic, there seems to be no difference between numerals and the concepts they refer to. But symbols do not generate the concepts that constitute their referents; they have to be linked to them by a thinking agent, even when this linkage has become automatic. It is, indeed, a ground rule of semiotics that a sound or a mark on paper becomes a symbol only when it is deliberately associated with a conceptual meaning.

I trust that you will agree that mathematics could not have happened if the concepts of‘unit’ and ‘Plurality of units’ had not somehow been generated. How this was done may not be quite so obyious.

Thinkers as diverse as Edmund Husserl, Albert Einstein, and Jean Piaget have stated very clearly that the concept of ‘unit’ is derived from the construction of ‘objects’ in our experiential world.

Einstein’s description of this construction is one of the clearest:

I believe that the first step in the setting of a ‘real external world’ is the formation of the concept of bodily objects and of bodily objects of various kinds. Out of the multitude of our sense experiences we take, mentally and arbitrarily, certain repeatedly occurring complexes of sense impressions (partly in conjunction with sense impressions which are interpreted as signs for sense experiences of others), and we correlate to them a concept — the concept of the bodily object. Considered logically this concept is not identical with the totality of sense impressions referred to; but it is a free creation of the human (or animal) mind. (Einstein, 1954, p. 291)

I have elsewhere shown that the concept of ‘plurality’, (unlike that of unitary physical object) cannot be derived from ‘sense impressions’, but only from the awareness that the recognition of a particular physical object is being repeated.

In Piaget’s terms, it is not an empirical abstraction from sensory-motor experience, but a reflective abstraction from the experiencer’s own mental operations. This is an important distinction. Let me try to make it as clear as possible. Initially, the operations of constructing or recognizing physical objects do require sense impressions; but the realization that one is carrying out the same recognition procedure more than once, arises from one’s own operating and not from the particular sensory material that furnishes the occasions for this way of operating.

‘Units’ and ‘pluralities’, however, do not yet constitute a basis for the development of mathematics or even arithmetic. At least one other concept is needed: the concept of ‘number’.

In our book on counting types, Steffe, Richards, Cobb, and I have presented a model of how an abstract concept of ‘Number’ may be derived from the activity of counting (Steffe et al, 1983). So far, I have not seen a more plausible model. But irrespective of the adequacy of that particular model, I have no doubt that both the notions of ordinal and cardinal number derive from the operations an active subject carries out and not from any specific sensory material.

If you agree that the concept of ‘Number’ requires both the concepts of ‘unit’ and ‘plurality’ — and if you further agree that without a concept of ‘number’ there can be no development of a mathematics of numbers, then it is clear that this mathematics is an affair of mental operations that have to be carried out by an active subject. As Hersh says:

Symbols are used as aids to thinking just as musical scores are used as aids to music. The music comes first, the score comes later. (Hersh, 1986, p. 19)

If this is so, it would seem to follow incontrovertibly that a string of mathematical symbols remains meaningless until someone has associated specific mental operations with the symbols. These operations, because they are mental operations, cannot be witnessed by anyone else. What can be witnessed, are the symbols that an acting subject produces in a spoken or written form as the result of his or her mental operations — and one can then examine whether or not they are compatible with the symbols one would have produced oneself.

This involves a two-fold process of interpretation:

• First, the other subject’s interpretation of the string of symbols he or she has produced;
• Second, one’s own interpretation of these as one perceives them.

I submit that this awareness of processes of interpretation would have far-reaching consequences for any theory of proof. But this is not my present concern.

Here, I want to emphasize that the conceptual analyses I have suggested would have far-reaching consequences also for the teaching of arithmetic. If mathematical symbols have to be interpreted in terms of mental operations, the teacher’s task is to stimulate and prod the student’s mind to operate mathematically. Sensory-motor material, graphic representations, and talk can provide occasions for the abstraction of mathematical operations, but they cannot convey them ready-made to the student. The frequently used expression ‘mathematical objects’ is misleading, because the meaning of mathematical symbols is never a static object but rather a particular way of operating. This goes for plurality and number, for point and line, and consequently for all the more complex abstractions of mathematics. The teacher’s task, then, is to orient the students’ mental operating; and to do this, the teacher needs at least a hypothetical model of how the student’s mind operates at the outset of the lesson (von Glasersfeld and Steffe, 1991).

To conclude, I would summarize these consequences by saying that, if we really want to teach arithmetic, we have to pay a great deal of attention to the mental operations of our students. Teaching has to be concerned with understanding rather than performance and the rote-learning of, say, the multiplication table, or training the mechanical performance of algorithms — because training is suitable only for animals whom one does not credit with a thinking mind.

References

Einstein, A. (1954) ‘Physics and reality’, Ideas and Opinions, New York, Bonanza Books.

Hersh, R. (1986) ‘Some proposals for revising the philosophy of mathematics’, in Tymoczko, T. (Ed) New Directions in the Philosophy of Mathematics (9–28), Boston/Basel, Birkhauser.

Steffe, L.P., von Glasersfeld, E. Richards, J. and Cobb, P. (1983) Children’s Counting Types: Philosophy, Theory, and Application, New York, Praeger, pp. 112–123.

von Glasersfeld, E. and Steffe, L.P. (1991) ‘Conceptual models in educational research and practice’, Journal of Educational Thought, 25, 2, pp. 91–103.

Chapter 2

Interaction and Children’s Mathematics

Leslie P. Steffe and Ron Tzur

Knowledge implies interaction, and we cannot step out of our domain of interactions, which is closed. We live, therefore, in a domain of subject-dependent knowledge and subject-dependent reality … We literally create the world in which we live by living it. (Maturana, 1978, pp. 60–1)

In recent years, the social interaction involved in the construction of the mathematics of children has been brought into the foreground in order to specify its constructive aspects (Bauersfeld, 1988; Yackel, et al., 1990). One of the basic goals in our current teaching experiment is to analyse such social interaction in the context of children working on fractions in computer micro worlds. In our analyses, however, we have found that social interaction does not provide a full account of children’s mathematical interaction. Children’s mathematical interaction also includes enactment or potential enactment of their operative mathematical schemes. Therefore, we conduct our analyses of children’s social interaction in the context o...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright
• Contents
• List of Tables and Figures
• Series Editor’s Preface
• Introduction
• Part 1: Constructivism and the Learning of Mathematics
• Part 2: Psychology, Epistemology and Hermeneutics
• Part 3: Enquiry in Mathematics Education
• Part 4: History, Mathematics and Education
• Notes on Contributors
• Index